Finite-difference scheme for one problem of nonlinear optics

Abstract

We consider a mathematical model, which describes Q-switching pro- cess. The finite difference scheme is developed for approximation of the given system of nonlinear PDEs. It is constructed by using the staggered grid, such a strategy enables an automatic linearization of the algorithm. The transport equations are approximated along characteristics z ±t, thus no discretization error is introduced at this stage. But such algorithm puts a strong relation between time and space steps of the discrete grid. The convergence analysis of this scheme is done using the method developed in [2]. First some estimates of the boundedness of the exact solution are proved. Then the boundedness of the discrete solution is investigated. On the basis of these estimates the main stability inequality is proved. The second order convergence rate with respect the space and time coordinates follows from this estimate.